Global Ill-Posedness of the Isentropic System of Gas Dynamics

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p(\rho)=\rho^2 and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions.